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G = C2×C6×C12order 144 = 24·32

Abelian group of type [2,6,12]

direct product, abelian, monomial

Aliases: C2×C6×C12, SmallGroup(144,178)

Series: Derived Chief Lower central Upper central

C1 — C2×C6×C12
C1C2C6C3×C6C3×C12C6×C12 — C2×C6×C12
C1 — C2×C6×C12
C1 — C2×C6×C12

Generators and relations for C2×C6×C12
 G = < a,b,c | a2=b6=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 162, all normal (8 characteristic)
C1, C2, C2 [×6], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C3×C6, C3×C6 [×6], C2×C12 [×24], C22×C6 [×4], C3×C12 [×4], C62 [×7], C22×C12 [×4], C6×C12 [×6], C2×C62, C2×C6×C12
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], C3×C12 [×4], C62 [×7], C22×C12 [×4], C6×C12 [×6], C2×C62, C2×C6×C12

Smallest permutation representation of C2×C6×C12
Regular action on 144 points
Generators in S144
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 97)(36 98)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 83)(50 84)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 121)(109 142)(110 143)(111 144)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 141)
(1 105 90 134 59 68)(2 106 91 135 60 69)(3 107 92 136 49 70)(4 108 93 137 50 71)(5 97 94 138 51 72)(6 98 95 139 52 61)(7 99 96 140 53 62)(8 100 85 141 54 63)(9 101 86 142 55 64)(10 102 87 143 56 65)(11 103 88 144 57 66)(12 104 89 133 58 67)(13 31 42 113 81 129)(14 32 43 114 82 130)(15 33 44 115 83 131)(16 34 45 116 84 132)(17 35 46 117 73 121)(18 36 47 118 74 122)(19 25 48 119 75 123)(20 26 37 120 76 124)(21 27 38 109 77 125)(22 28 39 110 78 126)(23 29 40 111 79 127)(24 30 41 112 80 128)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,83)(50,84)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(109,142)(110,143)(111,144)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141), (1,105,90,134,59,68)(2,106,91,135,60,69)(3,107,92,136,49,70)(4,108,93,137,50,71)(5,97,94,138,51,72)(6,98,95,139,52,61)(7,99,96,140,53,62)(8,100,85,141,54,63)(9,101,86,142,55,64)(10,102,87,143,56,65)(11,103,88,144,57,66)(12,104,89,133,58,67)(13,31,42,113,81,129)(14,32,43,114,82,130)(15,33,44,115,83,131)(16,34,45,116,84,132)(17,35,46,117,73,121)(18,36,47,118,74,122)(19,25,48,119,75,123)(20,26,37,120,76,124)(21,27,38,109,77,125)(22,28,39,110,78,126)(23,29,40,111,79,127)(24,30,41,112,80,128), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)>;

G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,97)(36,98)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,83)(50,84)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(109,142)(110,143)(111,144)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141), (1,105,90,134,59,68)(2,106,91,135,60,69)(3,107,92,136,49,70)(4,108,93,137,50,71)(5,97,94,138,51,72)(6,98,95,139,52,61)(7,99,96,140,53,62)(8,100,85,141,54,63)(9,101,86,142,55,64)(10,102,87,143,56,65)(11,103,88,144,57,66)(12,104,89,133,58,67)(13,31,42,113,81,129)(14,32,43,114,82,130)(15,33,44,115,83,131)(16,34,45,116,84,132)(17,35,46,117,73,121)(18,36,47,118,74,122)(19,25,48,119,75,123)(20,26,37,120,76,124)(21,27,38,109,77,125)(22,28,39,110,78,126)(23,29,40,111,79,127)(24,30,41,112,80,128), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,97),(36,98),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,83),(50,84),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,121),(109,142),(110,143),(111,144),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,141)], [(1,105,90,134,59,68),(2,106,91,135,60,69),(3,107,92,136,49,70),(4,108,93,137,50,71),(5,97,94,138,51,72),(6,98,95,139,52,61),(7,99,96,140,53,62),(8,100,85,141,54,63),(9,101,86,142,55,64),(10,102,87,143,56,65),(11,103,88,144,57,66),(12,104,89,133,58,67),(13,31,42,113,81,129),(14,32,43,114,82,130),(15,33,44,115,83,131),(16,34,45,116,84,132),(17,35,46,117,73,121),(18,36,47,118,74,122),(19,25,48,119,75,123),(20,26,37,120,76,124),(21,27,38,109,77,125),(22,28,39,110,78,126),(23,29,40,111,79,127),(24,30,41,112,80,128)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)])

C2×C6×C12 is a maximal subgroup of   C627C8  C62.15Q8  C6210Q8  C62.247C23  C62.129D4  C6219D4

144 conjugacy classes

class 1 2A···2G3A···3H4A···4H6A···6BD12A···12BL
order12···23···34···46···612···12
size11···11···11···11···11···1

144 irreducible representations

dim11111111
type+++
imageC1C2C2C3C4C6C6C12
kernelC2×C6×C12C6×C12C2×C62C22×C12C62C2×C12C22×C6C2×C6
# reps1618848864

Matrix representation of C2×C6×C12 in GL3(𝔽13) generated by

1200
0120
001
,
1200
0100
0012
,
800
040
009
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,1],[12,0,0,0,10,0,0,0,12],[8,0,0,0,4,0,0,0,9] >;

C2×C6×C12 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_{12}
% in TeX

G:=Group("C2xC6xC12");
// GroupNames label

G:=SmallGroup(144,178);
// by ID

G=gap.SmallGroup(144,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-2,432]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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